# Pranay GoswamiAmbedkar University Delhi | AUD · school of liberal studies

Pranay Goswami

Ph. D.

## About

83

Publications

18,305

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359

Citations

Introduction

Currently working at Ambedkar University Delhi as an assistant professor. My areas of interest are Geometric Function Theory, Fractional Calculus and Mathematical Modelling.

Additional affiliations

June 2012 - present

July 2008 - June 2012

**Amity University Rajasthan**

Position

- Lecturer

June 2006 - June 2008

Education

July 2006 - April 2011

July 2003 - June 2005

**S. S. Jain Subodh (P. G. ) College**

Field of study

- Mathematics

July 2000 - June 2003

**S.S. Jain Subodh P. G. College**

Field of study

- Mathematics

## Publications

Publications (83)

In this paper we investigate some extensions of sufficient conditions for meromorphic multivalent functions in the open unit disk to be meromorphic multivalent starlike and convex of order ˛. Our results unify and extend some starlikeness and convexity conditions for meromorphic multivalent functions obtained by Xu et al. [2], and some interesting...

Here we investigate a majorization problem involving starlike function of complex order belonging to a certain class defined by means of fractional derivatives. Relevant connections of the main results obtained in this paper with those given by earlier workers on the subject are also pointed out.

We studied convolution properties of spirallike, starlike and convex functions, and some special cases of the main results are also pointed out. Further, we also obtained inclusion and convolution properties for some new subclasses on p-valent functions defined by using the Dziok–Srivastava operator.

The purpose of the present article is to establish two theorems for a class of analytic univalent functions defined in the open unit disk. The results are proved by using techniques involving the principle of differential subordination. Connections of these theorems with Bazilevic˘ functions are considered and the main results are applied to obtain...

In the present paper, the fractional Burgers equation is considered in the (n + 1)-dimension with a nonlinear analytic term where the derivatives are Caputo fractional derivatives. An approximate analytical solution is obtained using the homotopy perturbation method. The concept is illustrated with examples.

The paper is concerned with the SIZR mathematical model for an outbreak of zombie infection with time-dependent infection rate. This class of the SIZR model involves equations that relate the susceptible S(t), the infected I(t), the zombie Z(t), and removed population R(t). The well poseness of the model is presented. The proposed model is then out...

This paper presents a new method to solve the local fractional partial differential equations (LFPDEs) describing fractal vehicular traffic flow. Firstly, the existence and uniqueness of solutions to LFPDEs were proved and then two schemes known as the basic method (BM) and modified local fractional variational iteration method (LFVIM) were develop...

In this paper, we introduce three new classes S k a M , N ; μ , R k a μ , and T k a θ of analytic functions defined by Fournier–Ruscheweyh integral operator. For these classes, we investigate the majorization problem. Furthermore, a number of new results are shown to follow upon specializing the parameters involved in our main results.

In this paper, we introduce a new subclass of analytic bi-univalent functions defined by using q-derivative operator. Further, we obtain both some initial and general coefficient bounds, and also Fekete-Szegö inequalities for bi-univalent functions that belong to this class. Our results extend and improve some known results as special cases. Mathem...

In this paper, we introduced a new fractional derivative operator based on Lonezo Hartely function, which is called G-function. With the help of the operator, we solved a fractional diffusion equations. Some applications related to the operator is also discussed as form of corollaries.

In this paper, we consider the time-fractional two-mode coupled Burgers equation with the Caputo fractional derivative. A modified homotopy perturbation method coupled with Laplace transform (He-Laplace method) is applied to find its approximate analytical solution. The method is to decompose the equation into a series of linear equations, which ca...

This paper presents a new method to solve the local fractional partial differential equations (LFPDEs) describing fractal vehicular traffic flow. Firstly, the existence and uniqueness of solutions to LFPDEs was proved and then two schemes known as the basic method (BM) and modified local fractional variational iteration method (LFVIM) were develope...

In this article, q-homotopy analysis transformation method (q-HATM) has been applied to solve fractional partial differential equations. The q-HATM is a well known method, which is the outcome of the conjunction of q-Homotopy analysis method and Laplace transform. Which provides the solution of such problems in a very easy manner. In our analysis,...

In this paper, we solve the n-Generalized KdV equation by local fractional homotopy analysis method (LFHAM). Further, we analyze the approximate solution in the form of non-differentiable generalized functions defined on Cantor sets. Some examples and special cases of the main results are also discussed.

The paper aims to extend the model of the Ebola virus in bats to the mathematical model of fractional order using Atangana- Baleanu derivative operator. A detailed proof for the existence, uniqueness, and stability of the solution for the fractional mathematical model is presented. A numerical approach is used to find the solution of the stated mod...

The main object of this paper is to investigate and determine a sufficient condition for q-starlikeness and q-convexity for functions which are associated with normalized Gauss hypergeometric function.

In this paper, we solve the [Formula: see text]-Generalized KdV equation by local fractional homotopy analysis method (LFHAM). Further, we analyze the approximate solution in the form of non-differentiable generalized functions defined on Cantor sets. Some examples and special cases of the main results are also discussed.

Diabetes is a burning issue in the whole world. It is the imbalance between body glucose and insulin. The study of this imbalance is very much needed from a research point of view. For this reason, Bergman gave an important model named-Bergman minimal model. In the present work, using Caputo-Fabrizio (CF) fractional derivative, we generalize Bergma...

The main object of this paper is to investigate and determine a sucient condition for q-starlike and q-convex functions which are associated with generalized conuent hypergeometric function. Keywords: Univalent functions, convex and q-convex functions, starlike and q-starlike functions, q-derivative operator, q-number, generalized conuent hypergeom...

In this paper fractional differential transform method is implemented for
modelling and solving system of the time fractional chemical engineering
equations. In this method the solution of the chemical reaction, reactor,
and concentration equations are considered as convergent series with easily
computable components. Also, the obtained solutions h...

In the paper, (n+1)-dimensional fractional M-Burgers equation with a force term in the sense of the Caputo fractional derivative is considered. We solve this equation using homotopy perturbation method (HPM) and find its approximate analytical solution. We illustrate the method with some concrete examples. We also provide a graphical representation...

In this paper, we extend the Burger's equation to the time-fractional Burger's equation based on different derivative operators as Yang-Abdel-Cattani, Atangana-Baleanu, Caputo-Fabrizio, and Liouville-Caputo. The analytical solutions for these different time-fractional Burger's equation are determined by employing the δ− Homotopy Analysis Transform...

This paper tends to investigate a few new subclasses of bi-univalent functions H(v, λ, n) as well as G(v, ξ, n) in an open disc U of unit length. We also appraises the coefficients |b2| & |b3| to Taylor-Maclaurin’s series.

In this paper fractional differential transform method is implemented for modelling and solving system of the time fractional chemical engineering equations. In this method the solution of the chemical reaction, reactor, and concentration equations are considered as convergent series with easily computable components. Also, the obtained solutions h...

Diabetes is one of the burning issues of the whole world. It effected the world population rapidly. According to the WHO approx 415 million people are living with diabetes in the world and this figure is expected to rise up to 642 million by 2040. World various organizations raise their voice against the dire facts about the increasing graph of dia...

Overview
A modern and in-depth presentation of classical complex analysis
Contains a large number of exercises with their complete solutions
Ideal for course adoption, but also for self-study.
Aims and Scope
This book is an in-depth and modern presentation of important classical results in complex analysis and is suitable for a first course on t...

A modern and in-depth presentation of classical complex analysis
Contains a large number of exercises with their complete solutions
Ideal for course adoption, but also for self-study
Aims and Scope
This book is an in-depth and modern presentation of important classical results in complex analysis and is suitable for a first course on the topic, as...

In this paper, several new integral inequalities are established by using Katugampola integral operator.

In the present paper, we give the best estimates for the norm of pre-Schwarzian derivatives
||T f(z)||
for subclasses of bi-univalent functions.

In the present paper, we derive the solution of the nonlinear fractional partial differential equations using an efficient approach based on the q-homotopy analysis transform method (q-HATM). The fractional diffusion equations derivatives are considered in Caputo sense. The derived results are graphically demonstrated as well.

In this paper, we introduce a unified subclass of analytic functions by making use of the principle of subordination, involving generalized Ruscheweyh Derivative operator Jλ,μp. The properties such as inclusion relationships, distortion theorems, coefficient inequalities and differential sandwich theorem for the above class have been discussed

In the present paper, we use analytical techniques to solve fractional nonlinear differential equations systems that arise in Bergman’s minimal model, used to describe blood glucose and insulin metabolism, after intravenous tolerance testing. We also discuss the stability and uniqueness of the solution.

In the present paper, we use analytical techniques to solve fractional nonlinear differential equations systems that arise in Bergman's minimal model, used to describe blood glucose and insulin metabolism, after intravenous tolerance testing. We also discuss the stability and uniqueness of the solution.

A nondifferentiable analytical solution of the n-dimensional diffusion equation in fractal heat transfer is investigated using the local fractional Adomian decomposition method.

In this paper with the help of Sumudu transform and variational iteration method will solve differential equations and successfully extended it for fractional differential equation related to entropy, wavelets etc. The process of the methods which produce solutions in terms of convergent series is explained. Some examples are provided to show the a...

We present iteration formulae of a fractional space-time telegraph equation using the combination of fractional variational iteration method and local fractional Laplace transform.

Applying the Faber polynomial expansions, we obtain the general coefficient bounds for the class of biunivalent functions with bounded boundary rotations.

In this chapter we solve generalized fractional diierential equations by using the Sumudu transform. The equations we treat include diierential equations with generalized Riemann-Liouville fractional derivatives. The operators involved are very general in nature, and cover a wide range of fractional diierential equations and their solutions in term...

We present the solution of generalized space time fractional telegraph equation by using Sumudu variational iteration method which is the combination of variational iteration method and Sumudu transform. We tried to overcome the difficulties in finding the value of Lagrange multiplier by this new technique.

In this paper, estimates for second and third MacLaurin coefficients of
certain subclasses of bi-univalent functions in the open unit disk defined by
convolution are determined, and certain special cases are also indicated. The
main result extends and improve a recent one obtained by Srivastava et al.

In this chapter we solve generalized fractional differential equations by using the Sumudu transform. The equations we treat include differential equations with generalized Riemann-Liouville fractional derivatives. The operators involved are very general in nature, and cover a wide range of fractional differential equations and their solutions in t...

Let \(\Omega\subset\mathbb{C}\), let p be analytic in the unit disc \(\mathrm{U}=\left\{z\in\mathbb{C}:|z|<1\right\}\), and let \(\psi(r,s,t;z):\mathbb{C}^3\times\mathrm{U}\rightarrow\mathbb{C}\). In a series of articles, S. S. Miller, P. T. Mocanu and many others have determined properties of functions ψ that satisfy the differential subordination...

In the present paper we obtain generalized sufficient conditions for the starlikeness and convexity of some normalized analytic functions; the results extend those recently obtained in the work of Uyanik and Owa (J. Inequal. Appl. 2011:87, 2011).
MSC:
30C45.

The purpose of the present paper is to introduce several new classes of p-valent meromorphic functions defined by Ruscheweyh derivative operator for meromorphic multivalent functions. Further, we investigate the radii problems of these analytic classes in the punctured unit disk.

We derive and discuss the analytical solution for the generalized time-fractional telegraph equation with the help of the Sumudu and Fourier transforms. In the process we use Green functions to derive the solution of the said differential equation.

In this paper we solve generalized fractional differential equations by using the Sumudu transform. The equations solved differential equations with generalized Riemann-Liouville fractional derivative operators are very general in nature and involve wide range of fractional differential equations and their solutions in terms of various functions re...

Recently M. K. Aouf and T. M. Seoudy, (2011, Integral Trans. Spec. Func. 22(6) (2011), 423-430) have introduced families of analytic functions associated with the Dziok-Srivastava operator. In this work we use the Dziok-Raina operator to consider classes of multivalent analytic functions. It is connected with Wright generalized hypergeometric funct...

In the present paper we introduce and studied two subclasses of
multivalent functions denoted by
$\mathcal{M}^{\lambda}_{p,n}(\gamma;\beta)$ and
$\mathcal{N}^{\lambda}_{p,n}(\mu,\eta;\delta)$. Further, by giving
specific values of the parameters of our main results, we will find some
connection between these two classes, and moreover, several conse...

We introduce a new integral operator for meromorphic multivalent functions. The starlikeness
condition of this integral operator is determined. Several special cases are
also discussed in the form of corollaries.

In this paper, we consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under a general integral operator defined by the Hadamard product.

In the present paper, we introduce a new integral operator for meromorphic functions. The starlikeness conditions of this integral operator are determined. Several special cases are also discussed in the form of corollaries.

The aim of this paper is to introduce two new classes
of analytic function by using principle of subordination and the Dziok-
Srivastava operator. We further investigate convolution properties for
these calsses. We also �nd necessary and su�cient condition and coe�-
cient estimate for them.

In the present paper, we give a sufficient condition to guarantee the solution of the constant coefficient fractional differential equation by the Sumudu transform.

We give a sufficient condition to guarantee the solution of the constant coefficient fractional differential equation by the Sumudu transform.

In the present paper, we study convolution properties for certain classes of meromophic multivalent functions. Further, with the help of convolution, coefficient estimates and inclusion relationship for these classes are also discussed.

In the present paper we introduce and investigate an interesting subclass
${\mathcal{K}_{s}^{(k)}(\lambda,h)}$
of analytic and close-to-convex functions in the open unit disk
${\mathbb{U}}$
. For functions belonging to the class
${\mathcal{K}_{s}^{(k)}(\lambda,h)}$
, we derive several properties as the inclusion relationships and distortion t...

In the present paper we obtain some conditions on a, b and
c to verify that zp
2F1(a; b; c; z) to be in various subclasses of
starlike and convex functions. we also examine an integral
operator related to the p-valent hypergeometric function.

We study some generalized integral operators for the classes of p-valent
functions with bounded radius and boundary rotation. Our work generalizes many
previously known results. Many of our results are best possible.

In the present paper, we investigate the majorization properties for certain classes of multivalent analytic functions defined by the Sălăgean operator. Moreover, we point out some new and interesting consequences of our main result.

In the present paper we investigate the majorization properties for certain classes of multivalent analytic functions defined by multiplier transformation. Moreover, we point out some new or known consequences of our main result.

We investigate familar geometric properties of the classes S p * [A,B], K p [A,B] and S p μ [A,B]. Also, results obtained earlier are derived as special cases from our main results.

We introduce a new subclass of meromorphic multivalent functions associated with Wright generalized hypergeometric functions, and obtain new results for this class by application of Briot-Bouquet differential subordination.

We investigate the majorization properties for certain classes of multivalent analytic functions defined by fractional derivaties. Moreover, we point out some new or known consequences of our main result.

We establish certain results concerning the quasi-Hadamard product for classes related to meromorphic p-valent analytic functions with positive coefficients.

In the present paper, we investigate majorization properties for the subclass of analytic functions
defined by an extension, introduced by Saitoh, of the well-known Carlson-Shaffer linear operator,
using differential subordination. Relevant connections of the main results obtained in this paper
with those given by earlier workers are also pointed o...

The purpose of the present paper is to investigate some argument properties for certain analytic functions in the open unit disk associated with the convolution structure. Some interesting applications are also considered as special cases of main results presented here.

In this paper we consider a class of analytic and multivalent functions to obtain some sucient conditions of starlikeness

We derive certain new argument properties of a class of multivalent analytic functions defined in the open unit disk by using a theorem recently established by A. Y. Lashin [JIPAM, J. Inequal. Pure Appl. Math. 5, No. 4, Paper No. 111, 5 p., electronic only (2004; Zbl 1086.30018)]. Certain interesting (known or new) results are derived in the form o...

We derive subordination and superordination results for a family of normalized analytic functions
in the open unit disk defined by integral operators. We apply this to obtain sandwich results and
generalizations of some known results.

Recently several research workers have obtained many interesting results involving the Dziok-Srivastava linear operator. In this paper we establish certain results for a subclass of multivalent functions involving this operator by using differential subordination. Relevant connections of our results with various (known or new) results useful in geo...

We establish certain results concerning the quasi-Hadamard product of certain analytic and p-valent functions with negative coefficients in the open unit disc.

## Questions

Question (1)

What is the union or intersection of two empty or nullset ?

## Projects

Projects (3)

Potential topics include but are not limited to the following:
Univalent and multivalent analytic functions
Planar harmonic mappings
Special functions and series
Differential subordination and superordination
Entire